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Technical Report: About FFT Analyzers 12

7. FFT and Time Window

7.1 Rectangular window

(Formula 7-2)

The rectangular pulse p(t) defined by is called the rectangular window (square wave window function). This is simply the same as the data cut off at time length T. The Fourier transform of this is from the previous section;

(Formula 7-3)

The function P (ω) can be represented graphically as shown in Figure 7-5. The upper part of the figure shows the shape of the time window, and the lower part shows the logarithmic representation of the absolute value of the spectrum, with the unit of the frequency axis scale being the reciprocal of the time window length.

  • Figure 7-5

Note that Figure 7-5 is displayed in a normalized state so that it becomes 1 when f is 0. This allows for a direct reading of the amplitude when f is 0. Please note that all subsequent figures are also displayed in a normalized state.

Adaptation to f(t) = cos ω 0 t

Let f(t) be cos ω 0 t. cos 2π f 0 t・ w(t) is a waveform in which the amplitude of the cosine waveform at frequency f 0 is changed by a factor of w(t) (amplitude modulated).

The Fourier transform of a waveform cut out through a square wave window is:

(Formula 7-4)

twist;

(Formula 7-5)

Adaptation to f(t) = sin ω 0 t

Similarly, consider a sine waveform instead of a cosine waveform:

(Formula 7-6)

The first and second terms of equations 7-5 and 7-6 are waveforms obtained by shifting the center 0 of P(ω) to ±f 0, and equation 7-5 is the sum of these, while equation 7-6 is the difference. Figure 7-6 shows the waveform of equation 7-6 when f 0 = 3/T. Now, the DFT values in Figure 7-6 are taken at the harmonic point with frequency 1/T, so looking at the value at that point:

When f = 0 (n = 0): W(f = 0) = 1

When f = 0 ± n /T (n = 1, 2, ...), then W(f) = 0

This clearly shows that sin ω 0 t has a frequency of f 0.

This is because, as shown in Figure 7-1 above, the segment is cut to match the period of the sine wave, resulting in a continuous start and end.

  • Figure 7-6

Next, consider the case where f1 = 3.5 / T is cut off without matching the period, as shown by the vertical line in Figure 7-7;

When f = 1 (n = 0): ≠ 0

When f 1 ± n / T (n = 1, 2, 3, .....): ≠ 0

The value is f 1 ± n / T. In DFT, there is no point for f 1, so unlike Figure 7-5, it will have a harmonic spectrum of f 0. This is an error caused by the discontinuity between the start and end points of the truncation, and is called a leakage error. When analyzing a signal that has a spectrum of f n in addition to f 0, if the f n component is too small, it will be buried in the leakage error of f 0. (See the rectangular window in Figure 8-3.)

  • Figure 7-7

Figure 7-8 shows Figures 7-6 and 7-7 superimposed on each other. As can be seen from this figure, the Y-axis value C at the point 3.5/T when f0 = 3/T is the same as the value B at 3/T when f1 = 3.5/T. Considering that the spectra other than B in Figure 7-7 are similar, we can consider the waveform at f1 = 3.5/T to have passed through a bandpass filter P(ω) (Figure 7-5 shifted to f0).

  • Figure 7-8

From the above, the concept of an FFT analyzer can be understood as DFT, which obtains the spectrum of the nth harmonic with f0 as the fundamental frequency, by replacing the bandpass filter, which has the shape of W(f), with a number of n bandpass filters arranged in a row, each with its center frequency shifted by f0. This is shown in Figure 7-9.

  • Figure 7-9

The rectangular window is a reasonable window for signals where f(t) is a percussive waveform with zero at the start and end points, but for continuous signals, leakage error results in poor frequency separation.

As we have seen, the change in the power spectrum due to the window function depends on the properties of the Fourier transform of the window function, so we only need to investigate the properties of W(f).

According to Percival's theorem, this power average is the same in both the frequency and time domains:

(Formula 7-7)

When normalized and expressed logarithmically, it becomes 0 dB, and its reciprocal is equal to the formula for the variance b. This represents the bandwidth assuming the same power and is one of the parameters that describe the window characteristics.